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cleartomark
%%EndFont 
%%BeginFont: CMSY6
%!PS-AdobeFont-1.1: CMSY6 1.0
%%CreationDate: 1991 Aug 15 07:21:34
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 48 /prime put
dup 76 /L put
readonly def
/FontBBox{-4 -948 1329 786}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI5
%!PS-AdobeFont-1.1: CMMI5 1.100
%%CreationDate: 1996 Aug 02 08:21:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI5 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 107 /k put
readonly def
/FontBBox{37 -250 1349 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI7
%!PS-AdobeFont-1.1: CMMI7 1.100
%%CreationDate: 1996 Jul 23 07:53:53
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 112 /p put
dup 113 /q put
readonly def
/FontBBox{0 -250 1171 750}readonly def
currentdict end
currentfile eexec
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532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470
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cleartomark
%%EndFont 
%%BeginFont: MSBM10
%!PS-AdobeFont-1.1: MSBM10 2.1
%%CreationDate: 1993 Sep 17 11:10:37
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 40 /subsetnoteql put
dup 69 /E put
dup 70 /F put
dup 78 /N put
dup 90 /Z put
dup 92 /hatwider put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 107 /k put
dup 109 /m put
dup 112 /p put
dup 113 /q put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR7
%!PS-AdobeFont-1.1: CMR7 1.0
%%CreationDate: 1991 Aug 20 16:39:21
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 48 /zero put
readonly def
/FontBBox{-27 -250 1122 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR6
%!PS-AdobeFont-1.1: CMR6 1.0
%%CreationDate: 1991 Aug 20 16:39:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 10 /Omega put
dup 40 /parenleft put
dup 41 /parenright put
dup 49 /one put
dup 50 /two put
dup 103 /g put
dup 108 /l put
dup 111 /o put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI6
%!PS-AdobeFont-1.1: CMMI6 1.100
%%CreationDate: 1996 Jul 23 07:53:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 73 /I put
dup 74 /J put
dup 82 /R put
dup 100 /d put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 109 /m put
dup 110 /n put
dup 114 /r put
dup 115 /s put
dup 116 /t put
readonly def
/FontBBox{11 -250 1241 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY8
%!PS-AdobeFont-1.1: CMSY8 1.0
%%CreationDate: 1991 Aug 15 07:22:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 3 /asteriskmath put
dup 48 /prime put
dup 50 /element put
dup 76 /L put
dup 80 /P put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMEX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /parenleftbig put
dup 1 /parenrightbig put
dup 2 /bracketleftbig put
dup 3 /bracketrightbig put
dup 12 /vextendsingle put
dup 16 /parenleftBig put
dup 17 /parenrightBig put
dup 18 /parenleftbigg put
dup 19 /parenrightbigg put
dup 20 /bracketleftbigg put
dup 21 /bracketrightbigg put
dup 26 /braceleftbigg put
dup 27 /bracerightbigg put
dup 80 /summationtext put
dup 88 /summationdisplay put
dup 89 /productdisplay put
dup 94 /logicalanddisplay put
dup 99 /hatwider put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 1 /Delta put
dup 10 /Omega put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 58 /colon put
dup 61 /equal put
dup 80 /P put
dup 91 /bracketleft put
dup 93 /bracketright put
dup 94 /circumflex put
dup 97 /a put
dup 100 /d put
dup 101 /e put
dup 103 /g put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 114 /r put
dup 120 /x put
readonly def
/FontBBox{-34 -251 988 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 1 /periodcentered put
dup 2 /multiply put
dup 3 /asteriskmath put
dup 8 /circleplus put
dup 14 /openbullet put
dup 17 /equivalence put
dup 18 /reflexsubset put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 26 /propersubset put
dup 33 /arrowright put
dup 50 /element put
dup 54 /negationslash put
dup 55 /mapsto put
dup 76 /L put
dup 80 /P put
dup 92 /intersection put
dup 94 /logicaland put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /alpha put
dup 12 /beta put
dup 13 /gamma put
dup 14 /delta put
dup 21 /lambda put
dup 22 /mu put
dup 25 /pi put
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 61 /slash put
dup 62 /greater put
dup 65 /A put
dup 67 /C put
dup 71 /G put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 79 /O put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 89 /Y put
dup 90 /Z put
dup 96 /lscript put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
currentdict end
currentfile eexec
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31B86E26A69791F1D910871DC7DDE8A0BE81C9A9F1A4D06EFA9D2E29471D1BCF
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01A3A3F4FFD0AEFD6DDC26C6E6298F268D7F5484F26DDE9DA77A15171030E056
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0FD34250638530B5B19999295391B63AACE876B91204E59BCC978255A38A13E2
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AAF694F630EAED470A154D02D13E15A5ACAD482DD58AB8E5DE0DA576EDF9E474
08DFFE933FC0E8DA18D2D29D1A01BACB0AFE4DC6668CFF45165E134F769AF095
9AEC420E9F08EAFBCFAFA82B5663DAD7CC7DED3206BF5957291DF837535DB23B
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cleartomark
%%EndFont 
%%BeginFont: CMSY7
%!PS-AdobeFont-1.1: CMSY7 1.0
%%CreationDate: 1991 Aug 15 07:21:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 3 /asteriskmath put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
currentdict end
currentfile eexec
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221A37D9A807DD01161779DDE7D251491EBF65A98C9FE2B1CF8D725A70281949
8F4AFFE638BBA6B12386C7F32BA350D62EA218D5B24EE612C2C20F43CD3BFD0D
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E611BE0A41A1F8C89C3BC16B352C3443AB6F665EAC5E0CC4229DECFC58E15765
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868AEA25DA3D5413467B76D2F02F8097D2841EDA6677731A6ACFEC0BABF1016A
089B2D24F83F8FEECD9B11A16D76AD6AD0934A8604FAE813A1304C950E76FC3E
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31FDD62F15E4D5BD0CC19D2E1C7364BCEE117028BAE01859A6AB353634EBC37C
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cleartomark
%%EndFont 
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 10 /Omega put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 54 /six put
dup 58 /colon put
dup 61 /equal put
dup 91 /bracketleft put
dup 93 /bracketright put
dup 103 /g put
dup 108 /l put
dup 111 /o put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
currentdict end
currentfile eexec
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211DE0E29718AEB1F22EFAB98670DE259EF61489758F88BA27F07109F68A0DDF
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4C0A48D5E94FB2CDC1031B5EF02D3E46FB372F0CBF27F9C01E0E755028C07A4E
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02C1580D2AB26E2C646D6879DDA9464FF7C3CC90814B9AA5FBC18167620C9C0D
18455D24BF9B517FC604EA90632C942FC461BEBE37ECAB8738B8C239FC141EE4
0D1F423E47A56DFB2BF2D65E50E269858F4C1D4C4425B6705E52A6E1C668D36D
378DAF10837B27E614BEFF2F79B6CA2FF88AB36D556AFB68182EDF0761607AA2
AC16519B933177431D7C68BACC587DD8BF61EA768AEE9DDF548807186D12699C
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cleartomark
%%EndFont 
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /alpha put
dup 12 /beta put
dup 13 /gamma put
dup 15 /epsilon1 put
dup 25 /pi put
dup 58 /period put
dup 59 /comma put
dup 61 /slash put
dup 65 /A put
dup 71 /G put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 79 /O put
dup 83 /S put
dup 89 /Y put
dup 96 /lscript put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 102 /f put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
currentdict end
currentfile eexec
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2537 y(in)m(teger)22 b Fw(m)h FA(is)e(de\034ned)j(to)d(b)s(e)i(the)f
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2537 y FA(.)40 b(As)23 b(is)e(the)i(con)m(v)m(en)m(tion,)i(w)m(e)257
2657 y(o)m(v)m(erload)h(these)i(terms)e(to)g(also)f(mean)g(the)i
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2778 y(on)41 b(size.)67 b(Understanding)41 b(the)f(computational)e
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y FA(is)g(a)g(ma)5 b(jor)257 2898 y(goal)31 b(of)h(computational)e
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3018 y(Smolesnky[)881 3018 y
SDict begin H.S end
 881 3018 a Black FA(40)p
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SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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b(in)g(the)h(late)f(eigh)m(ties,)k(building)46 b(up)s(on)j(the)g
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3139 y
SDict begin H.S end
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SDict begin H.R end
 836
3074 a 836 3139 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Razborov87) cvn H.B /ANN pdfmark end
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257 3760 y
SDict begin H.S end
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SDict begin 14.5 H.A end
 257 3760 a 257 3760
a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.1.2) cvn H.B /DEST pdfmark
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SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (Hfootnote.2) cvn H.B /DEST pdfmark
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SDict begin H.S end
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SDict begin H.S end
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SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.1.4) cvn H.B /DEST pdfmark
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SDict begin H.S end
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2273 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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a
SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.1.5) cvn H.B /DEST pdfmark
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SDict begin H.R end
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SDict begin H.S end
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SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (Hfootnote.3) cvn H.B /DEST pdfmark
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SDict begin H.S end
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a 994 5189 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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SDict begin H.S end
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SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (page.6) cvn H.B /DEST pdfmark
end
 257 266
a Black 404 573 a FA(While)39 b(it)g(is)h(immediate)e(that)i(gen-deg)
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b(it)e(is)f(a)257 693 y(cen)m(tral)45 b(question)g(in)e(the)i(theory)g
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Fw(m)p FA(,)j(it)c(seems)i(fairly)e(non-trivial)e(to)j(get)257
1054 y(go)s(o)s(d)33 b(estimates)g(of)h(deg)1186 1012
y FE(A)1186 1079 y(m)1252 1054 y Fr(\()p Fw(f)11 b Fr(\))34
b FA(for)f(ev)m(en)i(a)f(simple)e Fw(f)44 b FA(lik)m(e)33
b(OR)h(and)f(AND.)i(Ho)m(w)m(ev)m(er,)257 1175 y(when)g
Fw(m)f FA(is)f(a)g(prime)f Fw(p)h FA(\(or)g(a)g(prime)g(p)s(o)m(w)m
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1295 y(simple)e(and)h(elegan)m(t)g(fashion.)42 b(The)32
b(fact)f(that)g Ft(Z)2117 1259 y Fn(\003)2117 1320 y
FE(p)2188 1295 y FA(is)f(a)h(group)g(turns)h(out)f(to)f(b)s(e)i(v)m
(ery)257 1416 y(useful:)257 1416 y
SDict begin H.S end
 257 1416 a 257 1416
a
SDict begin 14.5 H.A end
 257 1416 a 257 1416 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.2) cvn H.B /DEST pdfmark
end
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Fw(Q)g FA(is)f(larger)g(than)h Fw(P)45 b FA(b)m(y)257
2554 y(a)33 b(factor)f(of)g(at)g(most)g Fw(p)22 b Fu(\000)h
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y
SDict begin H.S end
 295 2794 a Black FA(2.1)p Black 420 2730 a
SDict begin H.R end
 420 2730
a 420 2794 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (equation.2.1) cvn H.B /ANN pdfmark end
 420 2794 a FA(\)\):)257 2819 y
SDict begin H.S end
 257 2819
a 257 2819 a
SDict begin 14.5 H.A end
 257 2819 a 257 2819 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.3) cvn H.B /DEST pdfmark
end
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x Fv(F)-9 b(act)38 b(2.3.)p Black 41 w Fs(F)-7 b(or)34
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b Fr(\))p Fw(=)p Fr(\()p Fw(p)21 b Fu(\000)i Fr(1\))p
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b Fs(gen-de)-5 b(g)1984 3283 y FE(p)2024 3260 y Fr(\()p
Fs(AND)p Fr(\))28 b Fu(\025)2542 3192 y Fw(n)p 2461 3237
220 4 v 2461 3328 a(p)22 b Fu(\000)h Fr(1)2690 3260 y
Fw(:)404 3528 y FA(Unfortunately)-8 b(,)30 b(when)i Fw(m)f
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Fw(p)3285 4385 y Fz(1)3324 4370 y Fr(\))p Fw(;)17 b Fr(\()p
Fw(a)257 4491 y Fr(mo)s(d)34 b Fw(p)527 4506 y Fz(2)566
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b(Th)m(us,)34 b Fr(0)e FA(and)g Fr(1)g FA(in)f Ft(Z)3428
4506 y FE(m)257 4611 y FA(corresp)s(ond)j(to)e(tuples)h
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SDict begin H.S end
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4639 a
SDict begin 14.5 H.A end
 257 4639 a 257 4639 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.4) cvn H.B /DEST pdfmark
end
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b(act)55 b(2.4.)p Black 50 w Fs(L)-5 b(et)49 b Fw(m)54
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Fw(A)h Fr(=)g Fu(f)p Fr(0)p Fu(g)p Fs(\))e(has)h(no)257
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b(Applying)63 b(the)i(Chinese)g(Remaindering)c(Theorem,)72
b Fw(R)65 b FA(giv)m(es)g(rise)f(to)p Black Black eop
end
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SDict begin H.S end
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SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (page.7) cvn H.B /DEST pdfmark
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b(that)257 934 y Fw(R)q Fr(\(1\))d Fu(\021)h Fr(1)32
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1536 4 66 v 3429 1473 59 4 v 3429 1536 V 3487 1536 4
66 v 404 1728 a(F)-8 b(act)615 1728 y
SDict begin H.S end
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FA(2.4)p Black 740 1664 a
SDict begin H.R end
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SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.2.4) cvn H.B /ANN pdfmark end
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a 26 w FA(has)27 b(turned)f(out)g(to)g(b)s(e)h(somewhat)f(of)f(a)h
(serious)h(blo)m(w)f(to)f(pro)m(ving)h(lo)m(w)m(er)257
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2089 y(and)33 b(Rudic)m(h)g([)809 2089 y
SDict begin H.S end
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FA(6)p Black 857 2025 a
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.BBR94) cvn H.B /ANN pdfmark end
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a FA(].)257 2114 y
SDict begin H.S end
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SDict begin 14.5 H.A end
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2114 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.5) cvn H.B /DEST pdfmark
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(constructions)j(in)f(com)m(binatorics)e([)2462 3081
y
SDict begin H.S end
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SDict begin H.R end
 2560 3017
a 2560 3081 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Grolmusz00) cvn H.B /ANN pdfmark end
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SDict begin H.S end
 2614 3081
a Black FA(22)p Black 2712 3017 a
SDict begin H.R end
 2712 3017 a 2712 3081
a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Gopalan06) cvn H.B /ANN pdfmark end
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3202 y(in)k(obtaining)f(e\036cien)m(t)i(lo)s(cally)d(deco)s(dable)i(co)
s(des)h(b)m(y)h(Efremenk)m(o)f([)2838 3202 y
SDict begin H.S end
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a Black FA(20)p Black 2936 3138 a
SDict begin H.R end
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a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Efremenko09) cvn H.B /ANN pdfmark end
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1468 3322 y
SDict begin H.S end
 1468 3322 a Black FA(42)p Black 1565 3258
a
SDict begin H.R end
 1565 3258 a 1565 3322 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.TB98) cvn H.B /ANN pdfmark end
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(wing)d(lo)m(w)m(er)i(b)s(ound)h(on)g(the)257 3443 y(generalized)32
b(degree)i(of)e(the)h(OR)f(function.)257 3462 y
SDict begin H.S end
 257 3462
a 257 3462 a
SDict begin 14.5 H.A end
 257 3462 a 257 3462 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.6) cvn H.B /DEST pdfmark
end
 257 3462 a Black 176
x Fv(Theorem)38 b(2.6)g(\([)989 3638 y
SDict begin H.S end
 989 3638 a Black
Fv(42)p Black 1101 3573 a
SDict begin H.R end
 1101 3573 a 1101 3638 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.TB98) cvn H.B /ANN pdfmark end
 1101
3638 a Fv(]\).)p Black 42 w Fs(L)-5 b(et)36 b Fw(m)f
Fs(have)g Fw(t)29 b Fu(\025)g Fr(2)36 b Fs(distinct)f(prime)g(factors,)
g(and)g(let)g Fw(q)40 b Fs(b)-5 b(e)257 3758 y(the)39
b(smal)5 b(lest)37 b(maximal)g(prime)g(p)-5 b(ower)38
b(divisor)f(of)h Fw(m)p Fs(.)55 b(Then,)38 b(gen-de)-5
b(g)2979 3782 y FE(m)3045 3758 y Fr(\()p Fs(OR)p Fr(\))38
b Fs(is)g(at)257 3939 y(le)-5 b(ast)479 3828 y Fq(\020\020)652
3900 y Fz(1)p 608 3916 125 4 v 608 3973 a FE(q)r Fn(\000)p
Fz(1)764 3939 y Fu(\000)23 b Fw(o)p Fr(\(1\))1036 3828
y Fq(\021)1111 3939 y Fr(log)18 b Fw(n)1312 3828 y Fq(\021)1418
3823 y Fl(1)p 1382 3835 103 3 v 1382 3877 a Fm(t)p Fe(\000)p
Fl(1)1498 3939 y Fs(.)404 4155 y FA(The)36 b(ab)s(o)m(v)m(e)f(lo)m(w)m
(er)h(and)f(upp)s(er)g(b)s(ounds)h(on)f(the)g(degree)h(for)e(OR)h(and)g
(AND)h(has)257 4276 y(not)g(b)s(een)g(impro)m(v)m(ed)f(in)g(more)f
(than)i(ten)g(y)m(ears)g(and)g(it)e(is)h(an)g(imp)s(ortan)m(t)f(c)m
(hallenge)257 4396 y(to)29 b(narro)m(w)g(do)m(wn)h(the)f(gap)g(b)s(et)m
(w)m(een)i(them.)42 b(On)29 b(the)g(other)g(hand,)h(w)m(e)g(sp)s
(eculate)g(the)257 4516 y(follo)m(wing:)257 4536 y
SDict begin H.S end
 257
4536 a 257 4536 a
SDict begin 14.5 H.A end
 257 4536 a 257 4536 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.7) cvn H.B /DEST pdfmark
end
 257 4536 a Black
176 x Fv(Conjecture)50 b(2.7.)p Black 47 w Fs(L)-5 b(et)44
b Fw(P)58 b Fs(b)-5 b(e)44 b(a)g(multiline)-5 b(ar)43
b(p)-5 b(olynomial)43 b(of)h(de)-5 b(gr)g(e)g(e)44 b
Fw(d)g Fs(over)f Ft(Z)3398 4727 y FE(m)3465 4712 y Fs(.)257
4832 y(L)-5 b(et)58 b Fw(a)70 b Fu(2)g Ft(Z)771 4847
y FE(m)896 4832 y Fs(b)-5 b(e)57 b(such)g(that)h(ther)-5
b(e)57 b(exists)h(an)f Fw(x)2276 4847 y Fz(0)2385 4832
y Fu(2)70 b(f)p Fr(0)p Fw(;)17 b Fr(1)p Fu(g)2763 4796
y FE(n)2867 4832 y Fs(with)57 b Fw(P)14 b Fr(\()p Fw(x)3271
4847 y Fz(0)3310 4832 y Fr(\))70 b Fu(\021)257 4952 y
Fw(a)35 b Fs(\(mo)-5 b(d)34 b Fw(m)p Fs(\))q(.Then)f(the)i(numb)-5
b(er)34 b(of)g(p)-5 b(oints)34 b(in)h(the)f(cub)-5 b(e)35
b(at)f(which)g Fw(P)48 b Fs(evaluates)34 b(to)h Fw(a)257
5073 y Fs(is)g(at)g(le)-5 b(ast)35 b Fr(2)750 5036 y
FE(n)p Fn(\000)p FE(O)r Fz(\()p FE(d)967 5013 y Fm(t)994
5036 y Fz(\))1026 5073 y Fs(,)g(wher)-5 b(e)34 b Fw(m)28
b Fr(=)f Fw(p)1631 5088 y Fz(1)1687 5073 y Fu(\001)17
b(\001)g(\001)e Fw(p)1869 5088 y FE(t)1934 5073 y Fs(and)34
b(e)-5 b(ach)34 b Fw(p)2391 5088 y FE(i)2454 5073 y Fs(is)h(a)f
(distinct)h(prime.)404 5268 y FA(It)28 b(is)g(simple)f(to)h(v)m(erify)h
(that)f(this)g(conjecture)i(implies)c(that)i(for)g(suc)m(h)i
(square-free)257 5388 y Fw(m)p FA(,)37 b(gen-deg)731
5412 y FE(m)797 5388 y Fr(\()p FA(OR)p Fr(\))p Fw(;)c
FA(gen-deg)1406 5412 y FE(m)1473 5388 y Fr(\()p FA(AND)q
Fr(\))f(=)g Fw(n)1968 5352 y Fz(\012\(1)p FE(=t)p Fz(\))2175
5388 y FA(.)52 b(The)36 b(conjecture)g(ab)s(o)m(v)m(e)h(admits)p
Black Black eop end
%%Page: 8 8
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grestore}if}{pop}ifelse}{pop}ifelse}if end
 0 0 a Black 257 266 a
SDict begin H.S end
 257
266 a Black Black 257 266 a
SDict begin H.R end
 257 266 a 257 266 a
SDict begin [ /View [/XYZ H.V] /Dest (page.8) cvn H.B /DEST pdfmark
end
 257 266
a Black 307 x FA(a)42 b(natural)f(mo)s(di\034cation)f(to)h(comp)s
(osites)h(with)g(rep)s(eated)h(prime)d(factors.)73 b(W)-8
b(e)42 b(do)257 693 y(not)g(state)h(that)e(formally)e(to)j(k)m(eep)i
(the)e(discussion)g(simple)f(and)h(fo)s(cussed)h(on)f(the)257
814 y(essen)m(tial)i(problem)e(that)h(lies)f(ahead.)76
b(Before)43 b(w)m(e)i(end)f(this)e(section,)47 b(it)42
b(is)h(w)m(orth)257 934 y(men)m(tioning)33 b(that)h(the)h(ab)s(o)m(v)m
(e)g(conjecture)g(is)f(kno)m(wn)i(to)e(b)s(e)g(true)h(for)f(prime)f(mo)
s(duli)257 1054 y(\(see)48 b(for)f(example)f([)1054 1054
y
SDict begin H.S end
 1054 1054 a Black FA(5)p Black 1103 990 a
SDict begin H.R end
 1103 990
a 1103 1054 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Bar92) cvn H.B /ANN pdfmark end
 1103 1054 a FA(]\).)87 b(Using)47 b(Ramsey)g(Theory)-8
b(,)52 b(P\351ladeau)47 b(and)g(Th\351rien)h([)3371 1054
y
SDict begin H.S end
 3371 1054 a Black FA(38)p Black 3468 990 a
SDict begin H.R end
 3468 990
a 3468 1054 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.PT88) cvn H.B /ANN pdfmark end
 3468 1054 a FA(])257 1175 y(pro)m(v)m(e)38
b(a)f(result)f(that)g(easily)g(implies)e(this)j(conjecture)g(for)f
(arbitrary)g Fw(m)h FA(as)g(long)e(as)257 1295 y(the)e(degree)h
Fw(d)e FA(is)g(a)g(constan)m(t.)257 1512 y
SDict begin H.S end
 257 1512 a
257 1512 a
SDict begin 14.5 H.A end
 257 1512 a 257 1512 a
SDict begin [ /View [/XYZ H.V] /Dest (subsection.2.1) cvn H.B /DEST
pdfmark end
 257 1512 a 141 x Fd(2.1)131
b(Computing)43 b(MOD)1568 1671 y Fi(q)257 1860 y FA(The)36
b(adv)-5 b(an)m(tage)34 b(of)g(comp)s(osites)g(o)m(v)m(er)i(primes)d
(is)h(not)h(limited)c(to)j(computing)g(AND)257 1981 y(and)50
b(OR.)g(Among)f(other)h(things,)k(Bhatnagar)49 b(et.al.[)2396
1981 y
SDict begin H.S end
 2396 1981 a Black FA(19)p Black 2493 1916 a
SDict begin H.R end
 2493
1916 a 2493 1981 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.BGL06) cvn H.B /ANN pdfmark end
 2493 1981 a FA(])h(sho)m(w)m(ed)i(that)e(one)g(can)
257 2101 y(compute)d(the)f(THRESHOLD)1467 2116 y FE(k)1557
2101 y FA(function)g(b)m(y)h(p)s(olynomials)c(of)j(degree)h
Fw(O)s Fr(\()p Fw(n)3274 2065 y Fz(1)p FE(=t)p Fz(+)p
FE(\017)3457 2101 y Fr(\))257 2221 y FA(o)m(v)m(er)i
Ft(Z)547 2236 y FE(m)614 2221 y FA(,)j(if)47 b Fw(m)h
FA(has)h Fw(t)f FA(distinct)f(prime)g(factors)h(and)g
Fw(k)j FA(is)c(a)h(constan)m(t.)91 b(This)48 b(is)257
2342 y(a)j(generalization)d(of)i(the)h(upp)s(er)h(b)s(ound)e(due)i(to)e
(Barrington)f(et.al.)98 b(as)50 b(OR)h(is)257 2462 y(just)44
b(THRESHOLD)1080 2477 y Fz(1)1121 2462 y FA(.)76 b(Bhatnagar)42
b(et.al.)75 b(w)m(ondered)45 b(if)d(in)m(teresting)h(degree)h(up-)257
2582 y(p)s(er)g(b)s(ounds)g(could)f(b)s(e)h(pro)m(v)m(ed)h(for)e(the)h
(simple)e(function)h(MOD)2819 2597 y FE(q)2857 2582 y
FA(.)77 b(Hansen)44 b([)3343 2582 y
SDict begin H.S end
 3343 2582 a Black
FA(31)p Black 3441 2518 a
SDict begin H.R end
 3441 2518 a 3441 2582 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Hansen06) cvn H.B /ANN pdfmark end
 3441
2582 a FA(],)257 2703 y(dispro)m(ving)32 b(a)h(conjecture)g(of)f
(Bhatnagar)g(et.al.)43 b([)2153 2703 y
SDict begin H.S end
 2153 2703 a Black
FA(19)p Black 2250 2639 a
SDict begin H.R end
 2250 2639 a 2250 2703 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.BGL06) cvn H.B /ANN pdfmark end
 2250
2703 a FA(],)33 b(sho)m(w)m(ed)i(the)e(follo)m(wing:)257
2728 y
SDict begin H.S end
 257 2728 a 257 2728 a
SDict begin 14.5 H.A end
 257 2728 a 257 2728 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.8) cvn H.B /DEST pdfmark
end
 257
2728 a Black 234 x Fv(Theorem)i(2.8)h(\(Hansen\).)p Black
40 w Fs(Assume)d Fw(m)28 b Fr(=)f Fw(p)2056 2977 y Fz(1)2112
2962 y Fu(\001)17 b(\001)g(\001)e Fw(p)2294 2977 y FE(t)2357
2962 y Fs(and)33 b Fw(q)k Fs(ar)-5 b(e)33 b(c)-5 b(o-prime)31
b(satisfy-)257 3082 y(ing)36 b(the)h(fol)5 b(lowing)35
b(c)-5 b(ondition:)47 b(ther)-5 b(e)37 b(exists)f(p)-5
b(ositive)35 b(inte)-5 b(gers)36 b Fw(b)2741 3097 y Fz(1)2781
3082 y Fw(;)17 b(:)g(:)g(:)f(;)h(b)3041 3097 y FE(t)3108
3082 y Fs(such)36 b(that)257 3128 y Fq(P)363 3154 y FE(t)363
3232 y(i)p Fz(=1)518 3164 y(1)p 507 3180 57 4 v 507 3237
a FE(b)537 3247 y Fm(i)602 3203 y Fw(<)27 b Fr(1)35 b
Fs(and)f Fw(p)1027 3218 y FE(i)1083 3203 y Fu(\025)28
b Fw(q)t(b)1276 3218 y FE(i)1339 3203 y Fs(for)35 b(al)5
b(l)34 b Fw(i)p Fs(.)45 b(Then)34 b(gen-de)-5 b(g)2310
3226 y FE(m)2377 3203 y Fr(\()p Fs(MOD)2650 3218 y FE(q)2688
3203 y Fr(\))28 b(=)f Fw(O)s Fr(\()p Fw(n)3031 3167 y
Fz(1)p FE(=t)3131 3203 y Fr(\))p Fs(.)404 3462 y FA(T)-8
b(ardos)34 b(and)g(Barrington's)f([)1510 3462 y
SDict begin H.S end
 1510
3462 a Black FA(42)p Black 1608 3398 a
SDict begin H.R end
 1608 3398 a 1608
3462 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.TB98) cvn H.B /ANN pdfmark end
 1608 3462 a FA(])h(tec)m(hnique)h(can)f(b)s(e)g(adapted)g(\(see)
h(for)f(example)257 3582 y([)284 3582 y
SDict begin H.S end
 284 3582 a Black
FA(13)p Black 382 3518 a
SDict begin H.R end
 382 3518 a 382 3582 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.CH05) cvn H.B /ANN pdfmark end
 382 3582
a FA(]\))39 b(to)g(pro)m(v)m(e)h(an)f Fr(\012)1093 3502
y Fq(\000)1139 3582 y Fr(\(log)18 b Fw(n)p Fr(\))1416
3546 y Fz(1)p FE(=)p Fz(\()p FE(t)p Fn(\000)p Fz(1\))1661
3502 y Fq(\001)1746 3582 y FA(lo)m(w)m(er)39 b(b)s(ound)h(on)f(gen-deg)
2782 3606 y FE(m)2848 3582 y Fr(\()p FA(MOD)3126 3597
y FE(q)3164 3582 y Fr(\))p FA(.)63 b(Suc)m(h)257 3703
y(b)s(ounds)33 b(degrade)g(with)e(the)i(n)m(um)m(b)s(er)f(of)g
(distinct)f(prime)g(factors)h(of)g Fw(m)p FA(.)44 b(In)32
b(a)g(break-)257 3823 y(through)44 b(w)m(ork,)k(Bourgain)43
b([)1377 3823 y
SDict begin H.S end
 1377 3823 a Black FA(11)p Black 1474
3759 a
SDict begin H.R end
 1474 3759 a 1474 3823 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Bourgain05) cvn H.B /ANN pdfmark end
 1474 3823 a FA(])h(pro)m(v)m(ed)i(an)e
Fr(\012\(log)18 b Fw(n)p Fr(\))44 b FA(lo)m(w)m(er)g(b)s(ound)g(on)g
(the)g(gen-)257 3944 y(eralized)c(MOD)863 3959 y FE(m)929
3944 y FA(-degree)h(of)f(MOD)1632 3959 y FE(q)1670 3944
y FA(.)67 b(Bourgain's)40 b(metho)s(d)g(is)g(in)m(teresting)g(due)i(to)
257 4064 y(sev)m(eral)34 b(reasons.)45 b(First,)32 b(it)g(pro)m(v)m(es)
i(something)e(stronger,)h(sho)m(wing)g(that)g(the)g Fs(c)-5
b(orr)g(e-)257 4184 y(lation)44 b FA(b)s(et)m(w)m(een)38
b(the)f(b)s(o)s(olean)e(function)h(computed)h(b)m(y)g(a)f
(sub-logarithmic)d(degree)257 4305 y(p)s(olynomial)41
b(o)m(v)m(er)k Ft(Z)1059 4320 y FE(m)1126 4305 y FA(,)j(w.r.t.)79
b(an)44 b(accepting)g(set,)k(and)d(MOD)2734 4320 y FE(q)2817
4305 y FA(is)e(exp)s(onen)m(tially)257 4425 y(small.)e(Suc)m(h)33
b(a)f(correlation)e(b)s(ound)i(w)m(as)h(not)f(kno)m(w)h(ev)m(en)g(for)f
(p)s(olynomials)c(mo)s(dulo)257 4545 y(primes,)41 b(a)e(mo)s(del)e
(whic)m(h)j(one)g(t)m(ypically)e(assumes)i(w)m(e)h(understand)f(w)m
(ell.)63 b(The)40 b(re-)257 4666 y(sult,)34 b(v)m(ery)h(signi\034can)m
(tly)-8 b(,)33 b(impro)m(v)m(es)h(up)s(on)f(a)h(long)e(line)h(of)g(w)m
(ork)h(\(see,)i(for)d(example,)257 4786 y([)284 4786
y
SDict begin H.S end
 284 4786 a Black FA(21)p Black 382 4722 a
SDict begin H.R end
 382 4722
a 382 4786 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Goldmann95) cvn H.B /ANN pdfmark end
 382 4786 a FA(,)453 4786 y
SDict begin H.S end
 453 4786 a Black
FA(12)p Black 550 4722 a
SDict begin H.R end
 550 4722 a 550 4786 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.CGT96) cvn H.B /ANN pdfmark end
 550 4786
a FA(,)621 4786 y
SDict begin H.S end
 621 4786 a Black FA(24)p Black 718
4722 a
SDict begin H.R end
 718 4722 a 718 4786 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Green99) cvn H.B /ANN pdfmark end
 718 4786 a FA(,)789 4786
y
SDict begin H.S end
 789 4786 a Black FA(4)p Black 838 4722 a
SDict begin H.R end
 838 4722 a
838 4786 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.AB01) cvn H.B /ANN pdfmark end
 838 4786 a FA(,)908 4786 y
SDict begin H.S end
 908 4786 a Black
FA(26)p Black 1006 4722 a
SDict begin H.R end
 1006 4722 a 1006 4786 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Green04) cvn H.B /ANN pdfmark end
 1006
4786 a FA(]\).)77 b(Second,)47 b(Bourgain's)c(metho)s(d)g(b)s(oils)f
(do)m(wn)i(to)f(estimating)257 4907 y(certain)28 b(exp)s(onen)m(tial)g
(sums.)42 b(This)29 b(is)f(an)g(elemen)m(tary)g(but)h(p)s(o)m(w)m
(erful)f(tec)m(hnique)h(that)257 5027 y(has)41 b(spa)m(wned)h(more)e
(recen)m(t)h(progress)g([)1804 5027 y
SDict begin H.S end
 1804 5027 a Black
FA(15)p Black 1902 4963 a
SDict begin H.R end
 1902 4963 a 1902 5027 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.CGPT06) cvn H.B /ANN pdfmark end
 1902
5027 a FA(,)1969 5027 y
SDict begin H.S end
 1969 5027 a Black FA(30)p Black
2067 4963 a
SDict begin H.R end
 2067 4963 a 2067 5027 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Hansen06a) cvn H.B /ANN pdfmark end
 2067 5027 a FA(,)2134
5027 y
SDict begin H.S end
 2134 5027 a Black FA(18)p Black 2232 4963 a
SDict begin H.R end
 2232
4963 a 2232 5027 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.CW09) cvn H.B /ANN pdfmark end
 2232 5027 a FA(].)66 b(Due)42 b(to)e(its)f(imp)s
(ortance,)i(w)m(e)257 5147 y(include)g(a)h(pro)s(of)f(of)g(Bourgain's)g
(result.)71 b(Our)42 b(treatmen)m(t)f(follo)m(ws)g(that)g(of)g(Chat-)
257 5268 y(topadh)m(y)m(a)m(y)32 b([)756 5268 y
SDict begin H.S end
 756 5268
a Black FA(14)p Black 853 5204 a
SDict begin H.R end
 853 5204 a 853 5268
a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.C06) cvn H.B /ANN pdfmark end
 853 5268 a FA(,)910 5268 y
SDict begin H.S end
 910 5268 a Black FA(16)p
Black 1007 5204 a
SDict begin H.R end
 1007 5204 a 1007 5268 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.C07) cvn H.B /ANN pdfmark end
 1007 5268 a
FA(],)f(that)e(is)h(v)m(ery)h(close)e(to)h(the)g(metho)s(d)f(of)g([)
2606 5268 y
SDict begin H.S end
 2606 5268 a Black FA(11)p Black 2704 5204
a
SDict begin H.R end
 2704 5204 a 2704 5268 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Bourgain05) cvn H.B /ANN pdfmark end
 2704 5268 a FA(,)2760 5268
y
SDict begin H.S end
 2760 5268 a Black FA(27)p Black 2858 5204 a
SDict begin H.R end
 2858 5204
a 2858 5268 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.GRS05) cvn H.B /ANN pdfmark end
 2858 5268 a FA(])h(but)g(is)f(sligh)m(tly)257
5388 y(simpler)i(and)i(sharp)s(er.)257 5407 y
SDict begin H.S end
 257 5407
a 257 5407 a
SDict begin 14.5 H.A end
 257 5407 a 257 5407 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.9) cvn H.B /DEST pdfmark
end
 257 5407 a Black Black
eop end
%%Page: 9 9
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 0 0 a Black 257 266 a
SDict begin H.S end
 257
266 a Black Black 257 266 a
SDict begin H.R end
 257 266 a 257 266 a
SDict begin [ /View [/XYZ H.V] /Dest (page.9) cvn H.B /DEST pdfmark
end
 257 266
a Black Black 307 x Fv(De\034nition)46 b(2.9.)p Black
46 w FA(F)-8 b(or)41 b(an)m(y)g Fw(b)i Fu(2)g(f)p Fr(0)p
Fw(;)17 b(:)g(:)g(:)e(;)i(q)32 b Fu(\000)c Fr(1)p Fu(g)p
FA(,)43 b(de\034ne)f(the)g Fw(b)p FA(th)f(MOD)3126 588
y FE(q)3164 573 y FA(-residue)257 693 y(class)33 b(of)f
Fu(f)p Fr(0)p Fw(;)17 b Fr(1)p Fu(g)839 657 y FE(n)885
693 y FA(,)33 b(denoted)g(b)m(y)g Fw(M)1542 708 y FE(q)1581
693 y Fr(\()p Fw(b)p Fr(\))p FA(,)g(as)g(the)f(follo)m(wing:)552
992 y Fw(M)646 1007 y FE(q)684 992 y Fr(\()p Fw(b)p Fr(\))c(=)g
Fu(f)p Fw(x)g Fr(=)f(\()p Fw(x)1262 1007 y Fz(1)1302
992 y Fw(;)17 b(:)g(:)g(:)f(;)h(x)1576 1007 y FE(n)1623
992 y Fr(\))28 b Fu(2)g(f)p Fr(0)p Fw(;)17 b Fr(1)p Fu(g)2025
951 y FE(n)2093 992 y Fu(j)2210 868 y FE(n)2159 898 y
Fq(X)2174 1108 y FE(i)p Fz(=1)2320 992 y Fw(x)2375 1007
y FE(i)2431 992 y Fr(=)27 b Fw(b)33 b FA(\(mo)s(d)f Fw(q)t
FA(\))p Fu(g)3295 992 y
SDict begin H.S end
 3295 992 a 3295 992 a
SDict begin 17.5 H.A end
 3295 992
a 3295 992 a
SDict begin [ /View [/XYZ H.V] /Dest (equation.2.2) cvn H.B /DEST pdfmark
end
 3295 992 a FA(\(2.2\))257 1218 y
SDict begin H.S end
 257 1218
a 257 1218 a
SDict begin 14.5 H.A end
 257 1218 a 257 1218 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.10) cvn H.B /DEST pdfmark
end
 257 1218 a Black 87
x Fv(De\034nition)45 b(2.10.)p Black 45 w FA(F)-8 b(or)39
b(an)m(y)i(p)s(olynomial)36 b Fw(P)53 b FA(o)m(v)m(er)41
b Ft(Z)2347 1320 y FE(m)2453 1305 y FA(and)f Fw(a)g Fu(2)h
Ft(Z)2914 1320 y FE(m)2981 1305 y FA(,)g(let)e Fw(P)3274
1269 y Fn(\000)p Fz(1)3368 1305 y Fr(\()p Fw(a)p Fr(\))257
1426 y FA(de\034ne)34 b(the)f(set)g(of)f(p)s(oin)m(ts)h(in)e
Fu(f)p Fr(0)p Fw(;)17 b Fr(1)p Fu(g)1619 1389 y FE(n)1698
1426 y FA(where)33 b Fw(P)46 b FA(ev)-5 b(aluates)33
b(to)f Fw(a)p FA(.)257 1630 y(An)24 b(in)m(tuition)c(ab)s(out)j(a)f
(random)g(and)h(uniform)e(set)j(is)f(that)f(eac)m(h)i(of)f(the)g
Fw(M)3018 1645 y FE(q)3056 1630 y Fr(\()p Fw(b)p Fr(\))h
FA(residue)257 1750 y(classes)43 b(are)f(equally)g(represen)m(ted)i(in)
d(suc)m(h)j(a)d(set.)73 b(Bourgain's)41 b(result)h(essen)m(tially)257
1870 y(sho)m(ws)35 b(that)e(if)g Fw(P)46 b FA(has)34
b(lo)m(w)f(degree,)h(then)g Fw(P)1933 1834 y Fn(\000)p
Fz(1)2027 1870 y Fr(\()p Fw(a)p Fr(\))f FA(app)s(ears)h(pseudorandom)
3167 1870 y
SDict begin H.S end
 3167 1870 a -36 x Fp(4)3207 1870 y
SDict begin 14.5 H.L end
 3207 1870
a 3207 1870 a
SDict begin [ /Subtype /Link /Dest (Hfootnote.5) cvn /H /I /Border
[0 0 12] /Color [1 0 0] H.B /ANN pdfmark end
 3207 1870 a 33 w FA(to)f(the)257 1991 y(MOD)497
2006 y FE(q)565 1991 y FA(function.)42 b(In)30 b(other)f(w)m(ords,)j
(either)d(eac)m(h)i(of)e(the)h(MOD)2662 2006 y FE(q)2729
1991 y FA(residue)h(classes)f(are)257 2111 y(almost)c(equally)h
(represen)m(ted)j(in)d Fw(P)1597 2075 y Fn(\000)p Fz(1)1691
2111 y Fr(\()p Fw(a)p Fr(\))g FA(or)g(the)h(set)h(is)e(a)g(v)m(ery)i
(small)c(fraction)i(of)g(the)257 2231 y(cub)s(e.)257
2231 y
SDict begin H.S end
 257 2231 a 257 2231 a
SDict begin 14.5 H.A end
 257 2231 a 257 2231 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.11) cvn H.B /DEST pdfmark
end
 257
2231 a Black 204 x Fv(Lemma)59 b(2.11)h(\(Bourgain's)f(Uniformit)m(y)e
(Lemma\).)p Black 50 w Fs(F)-7 b(or)52 b(al)5 b(l)52
b(p)-5 b(ositive)52 b(c)-5 b(o-)257 2556 y(prime)38 b(inte)-5
b(gers)37 b Fw(m;)17 b(q)t Fs(,)38 b(ther)-5 b(e)38 b(exists)g(a)g(p)-5
b(ositive)37 b(c)-5 b(onstant)37 b Fw(\015)i Fr(=)33
b Fw(\015)5 b Fr(\()p Fw(q)t Fr(\))33 b Fw(<)g Fr(1)38
b Fs(such)g(that)257 2676 y(for)c(every)g(p)-5 b(olynomial)33
b Fw(P)48 b Fs(of)34 b(de)-5 b(gr)g(e)g(e)33 b Fw(d)h
Fs(over)g Ft(Z)2039 2691 y FE(m)2140 2676 y Fs(and)f(every)h
Fw(a)28 b Fu(2)g Ft(Z)2824 2691 y FE(m)2891 2676 y Fs(,)35
b(the)f(fol)5 b(lowing)257 2797 y(holds:)321 3025 y Fq(\014)321
3085 y(\014)321 3145 y(\014)321 3205 y(\014)371 3170
y Fr(Pr)492 3089 y Fq(\002)533 3170 y Fw(x)29 b Fu(2)711
3089 y Fq(\000)756 3170 y Fw(P)833 3129 y Fn(\000)p Fz(1)927
3170 y Fr(\()p Fw(a)p Fr(\))22 b Fu(\\)h Fw(M)1259 3185
y FE(q)1297 3170 y Fr(\()p Fw(b)p Fr(\))1414 3089 y Fq(\001\003)1524
3170 y Fu(\000)1633 3102 y Fr(1)p 1633 3147 49 4 v 1634
3238 a Fw(q)1709 3170 y Fr(Pr)1830 3089 y Fq(\002)1871
3170 y Fw(x)28 b Fu(2)g Fw(P)2125 3129 y Fn(\000)p Fz(1)2219
3170 y Fr(\()p Fw(a)p Fr(\))2346 3089 y Fq(\003)2388
3025 y(\014)2388 3085 y(\014)2388 3145 y(\014)2388 3205
y(\014)2449 3170 y Fu(\024)g Fs(exp)2693 3029 y Fq(\022)2789
3170 y Fu(\000)3053 3102 y Fw(\015)5 b(n)p 2898 3147
423 4 v 2898 3183 a Fq(\000)2944 3263 y Fw(m)p Fr(2)3078
3235 y FE(m)p Fn(\000)p Fz(1)3235 3183 y Fq(\001)3281
3205 y FE(d)3331 3029 y Fq(\023)3404 3170 y Fw(:)3295
3391 y
SDict begin H.S end
 3295 3391 a 3295 3391 a
SDict begin 17.5 H.A end
 3295 3391 a 3295 3391
a
SDict begin [ /View [/XYZ H.V] /Dest (equation.2.3) cvn H.B /DEST pdfmark
end
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(pro)s(of,)i(let)d(us)h(recall)d(an)j(elemen)m(tary)f(fact)g(ab)s(out)g
(the)257 3724 y(primitiv)m(e)30 b(ro)s(ots)h(of)g(unit)m(y)h(that)g(w)m
(e)h(mak)m(e)f(rep)s(eated)g(use)h(of)e(henceforth.)44
b(Let)32 b Fr(e)3301 3739 y FE(m)3368 3724 y Fr(\()p
Fw(y)t Fr(\))257 3845 y FA(denote)24 b(the)e(primitiv)m(e)e
Fw(m)p FA(-th)j(ro)s(ot)e(of)h(unit)m(y)h(raised)f(to)g(the)h
Fw(y)t FA(th)e(p)s(o)m(w)m(er,)26 b(i.e.)40 b(exp)q Fr(\()3273
3801 y Fz(2)p FE(\031)r(j)t(y)p 3273 3822 148 4 v 3315
3879 a(m)3430 3845 y Fr(\))p FA(,)257 3965 y(where)34
b Fw(j)39 b FA(is)32 b(the)h(complex)f(square)h(ro)s(ot)f(of)g
Fu(\000)p Fr(1)p FA(.)44 b(Then,)257 3985 y
SDict begin H.S end
 257 3985
a 257 3985 a
SDict begin 14.5 H.A end
 257 3985 a 257 3985 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.12) cvn H.B /DEST pdfmark
end
 257 3985 a Black 184
x Fv(F)-9 b(act)35 b(2.12.)p Black 41 w Fs(If)d Fw(y)f
Fr(=)d(0)k Fs(then,)1372 4130 y Fz(1)p 1358 4146 63 4
v 1358 4204 a FE(m)1447 4094 y Fq(P)1552 4121 y FE(m)p
Fn(\000)p Fz(1)1552 4198 y FE(\013)p Fz(=0)1726 4169
y Fr(e)1769 4184 y FE(m)1835 4169 y Fr(\()p Fw(\013)q(y)t
Fr(\))g Fs(is)h Fr(1)g Fs(and)f(the)h(expr)-5 b(ession)32
b(is)h Fr(0)f Fs(other-)257 4290 y(wise.)257 4494 y FA(Armed)h(with)f
(this)g(basic)g(fact,)h(w)m(e)g(pro)m(v)m(e)h(the)f(Uniformit)m(y)e
(Lemma)g(b)s(elo)m(w:)p Black 257 4690 a Fs(Pr)-5 b(o)g(of)35
b(of)f(Uniformity)h(L)-5 b(emma.)p Black 48 w FA(W)d(e)34
b(write)f Fr(Pr)2064 4609 y Fq(\002)2105 4690 y Fw(x)d
Fu(2)2285 4609 y Fq(\000)2331 4690 y Fw(P)2408 4654 y
Fn(\000)p Fz(1)2502 4690 y Fr(\()p Fw(a)p Fr(\))23 b
Fu(\\)g Fw(M)2835 4705 y FE(q)2873 4690 y Fr(\()p Fw(b)p
Fr(\))2990 4609 y Fq(\001\003)3111 4690 y FA(as)34 b(an)f(ex-)257
4810 y(p)s(onen)m(tial)e(sum.)44 b(Th)m(us,)p Black 257
4898 1296 4 v 370 4959 a Fo(4)407 4990 y
SDict begin H.S end
 407 4990 a Black
Black 407 4990 a
SDict begin H.R end
 407 4990 a 407 4990 a
SDict begin [ /View [/XYZ H.V] /Dest (Hfootnote.5) cvn H.B /DEST pdfmark
end
 407 4990 a Fx(The)32
b(metho)r(d)g(emplo)n(y)n(ed)f(b)n(y)g(Bourgain)f(to)i(pro)n(v)n(e)e
(this)i(result)g(is)f(closely)g(related)g(to)h(metho)r(d)257
5089 y(emplo)n(y)n(ed)27 b(commonly)g(in)h(comm)n(unication)f
(complexit)n(y)g(to)h(estimate)f(the)i(discrepancy)d(of)i(a)f(func-)257
5189 y(tion.)64 b(Indeed,)39 b(the)e(quan)n(tit)n(y)e(in)i(the)g(LHS)g
(of)42 b(\()1879 5189 y
SDict begin H.S end
 1879 5189 a Black Fx(2.3)p Black
1986 5136 a
SDict begin H.R end
 1986 5136 a 1986 5189 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (equation.2.3) cvn H.B /ANN pdfmark end
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b(is)f(closely)f(related)h(to)g(the)h(discrepancy)e(of)257
5289 y(MOD)461 5301 y Fg(q)529 5289 y Fx(function)d(w.r.t.)47
b(p)r(olynomial)31 b(mappings)f(mo)r(dulo)h Fi(m)g Fx(.)48
b(The)31 b(in)n(terested)g(reader)e(can)i(\034nd)257
5388 y(more)c(details)g(on)h(this)g(p)r(oin)n(t)f(of)h(view)f(in)h([)
1628 5388 y
SDict begin H.S end
 1628 5388 a Black Fx(16)p Black 1711 5335
a
SDict begin H.R end
 1711 5335 a 1711 5388 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.C07) cvn H.B /ANN pdfmark end
 1711 5388 a Fx(,)1762 5388
y
SDict begin H.S end
 1762 5388 a Black Fx(17)p Black 1845 5335 a
SDict begin H.R end
 1845 5335
a 1845 5388 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.C08) cvn H.B /ANN pdfmark end
 1845 5388 a Fx(])p Black Black Black eop
end
%%Page: 10 10
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SDict begin /product where{pop product(Distiller)search{pop pop pop
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SDict begin H.S end
 257
266 a Black Black 257 266 a
SDict begin H.R end
 257 266 a 257 266 a
SDict begin [ /View [/XYZ H.V] /Dest (page.10) cvn H.B /DEST pdfmark
end
 257 266
a Black 274 805 a Fr(Pr)306 864 y FE(x)395 724 y Fq(\002)436
805 y Fw(x)29 b Fu(2)614 724 y Fq(\000)659 805 y Fw(P)736
763 y Fn(\000)p Fz(1)830 805 y Fr(\()p Fw(a)p Fr(\))22
b Fu(\\)h Fw(M)1162 820 y FE(q)1200 805 y Fr(\()p Fw(b)p
Fr(\))1317 724 y Fq(\001\003)285 1089 y Fr(=)28 b Ft(E)455
1104 y FE(x)p Fn(2f)p Fz(0)p FE(;)p Fz(1)p Fn(g)702 1085
y Fm(n)749 948 y Fq(\024\022)903 1021 y Fr(1)p 885 1066
86 4 v 885 1157 a Fw(m)997 964 y FE(m)p Fn(\000)p Fz(1)1001
994 y Fq(X)1005 1203 y FE(\013)p Fz(=0)1166 1089 y Fr(e)1209
1104 y FE(m)1276 1008 y Fq(\000)1322 1089 y Fw(\013)q
Fr(\()p Fw(P)14 b Fr(\()p Fw(x)p Fr(\))21 b Fu(\000)i
Fw(a)p Fr(\))1841 1008 y Fq(\001)1887 948 y(\023\022)2043
1021 y Fr(1)p 2043 1066 49 4 v 2044 1157 a Fw(q)2129
960 y FE(q)r Fn(\000)p Fz(1)2119 994 y Fq(X)2124 1206
y FE(\014)s Fz(=0)2279 1089 y Fr(e)2322 1104 y FE(q)2361
1008 y Fq(\000)2406 1089 y Fw(\014)6 b Fr(\()p Fw(x)2560
1104 y Fz(1)2622 1089 y Fr(+)22 b Fu(\001)17 b(\001)g(\001)j
Fr(+)i Fw(x)3011 1104 y FE(n)3081 1089 y Fu(\000)g Fw(b)p
Fr(\))3259 1008 y Fq(\001)3305 948 y(\023)q(\025)3295
1322 y
SDict begin H.S end
 3295 1322 a 3295 1322 a
SDict begin 17.5 H.A end
 3295 1322 a 3295 1322
a
SDict begin [ /View [/XYZ H.V] /Dest (equation.2.4) cvn H.B /DEST pdfmark
end
 3295 1322 a FA(\(2.4\))404 1550 y(Expanding)32 b(the)h(sum)g(inside)f
(the)g(second)i(m)m(ultiplicand)29 b(and)k(treating)e(the)i(case)257
1670 y(of)f Fw(\014)i Fr(=)27 b(0)33 b FA(separately)-8
b(,)32 b(one)h(gets)439 1987 y(\()477 1987 y
SDict begin H.S end
 477 1987
a Black FA(2.4)p Black 601 1922 a
SDict begin H.R end
 601 1922 a 601 1987
a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (equation.2.4) cvn H.B /ANN pdfmark end
 601 1987 a FA(\))28 b Fr(=)780 1919 y(1)p 780 1964
49 4 v 781 2055 a Fw(q)856 1987 y Ft(E)922 2002 y FE(x)966
1846 y Fq(\024)1047 1919 y Fr(1)p 1029 1964 86 4 v 1029
2055 a Fw(m)1140 1862 y FE(m)p Fn(\000)p Fz(1)1145 1892
y Fq(X)1149 2101 y FE(\013)p Fz(=0)1310 1987 y Fr(e)1353
2002 y FE(m)1420 1906 y Fq(\000)1465 1987 y Fw(\013)q
Fr(\()p Fw(P)14 b Fr(\()p Fw(x)p Fr(\))22 b Fu(\000)g
Fw(a)p Fr(\))1984 1906 y Fq(\001)2030 1846 y(\025)793
2276 y Fr(+)998 2209 y(1)p 956 2253 133 4 v 956 2344
a Fw(mq)1312 2181 y Fq(X)1115 2397 y FE(\013)p Fn(2)p
Fz([)p FE(m)p Fz(])p FE(;\014)s Fn(2)p Fz([)p FE(q)r
Fz(])p Fn(\000f)p Fz(0)p Fn(g)1669 2276 y Fw(S)1735 2235
y FE(m;q)1855 2276 y Fr(\()p Fw(\013)q(;)17 b(\014)6
b(;)17 b(P)d Fr(\)e)2263 2291 y FE(m)2328 2276 y Fr(\()p
Fu(\000)p Fw(a\013)q Fr(\)e)2638 2291 y FE(q)2677 2276
y Fr(\()p Fu(\000)p Fw(b\014)6 b Fr(\))3295 2276 y
SDict begin H.S end
 3295
2276 a 3295 2276 a
SDict begin 17.5 H.A end
 3295 2276 a 3295 2276 a
SDict begin [ /View [/XYZ H.V] /Dest (equation.2.5) cvn H.B /DEST pdfmark
end
 3295 2276
a FA(\(2.5\))404 2616 y(where,)500 3004 y Fw(S)566 2963
y FE(m;q)686 3004 y Fr(\()p Fw(\013)q(;)17 b(\014)6 b(;)17
b(P)d Fr(\))25 b(=)j Ft(E)1246 3020 y FE(x)p Fn(2f)p
Fz(0)p FE(;)p Fz(1)p Fn(g)1493 3001 y Fm(n)1540 2864
y Fq(\024)1593 3004 y Fr(e)1636 3019 y FE(m)1703 2923
y Fq(\000)1749 3004 y Fw(\013)q(P)14 b Fr(\()p Fw(x)p
Fr(\))2020 2923 y Fq(\001)2087 3004 y Fu(\001)22 b Fr(e)2180
3019 y FE(q)2218 2923 y Fq(\000)2264 3004 y Fw(\014)6
b Fr(\()p Fw(x)2418 3019 y Fz(1)2479 3004 y Fr(+)22 b
Fu(\001)17 b(\001)g(\001)k Fr(+)h Fw(x)2869 3019 y FE(n)2916
3004 y Fr(\))2954 2923 y Fq(\001)3000 2864 y(\025)3295
3004 y
SDict begin H.S end
 3295 3004 a 3295 3004 a
SDict begin 17.5 H.A end
 3295 3004 a 3295 3004
a
SDict begin [ /View [/XYZ H.V] /Dest (equation.2.6) cvn H.B /DEST pdfmark
end
 3295 3004 a FA(\(2.6\))404 3307 y(Observing)41 b(that)g(the)h
(\034rst)g(term)e(in)h(\()1885 3307 y
SDict begin H.S end
 1885 3307 a Black
FA(2.5)p Black 2010 3243 a
SDict begin H.R end
 2010 3243 a 2010 3307 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (equation.2.5) cvn H.B /ANN pdfmark end
 2010
3307 a FA(\))g(is)f(simply)2526 3268 y Fz(1)p 2526 3284
36 4 v 2527 3341 a FE(q)2588 3307 y Fr(Pr)2709 3226 y
Fq(\002)2751 3307 y Fw(x)j Fu(2)g Fw(P)3035 3271 y Fn(\000)p
Fz(1)3128 3307 y Fr(\()p Fw(a)p Fr(\))3255 3226 y Fq(\003)3338
3307 y FA(and)257 3437 y Fu(j)p Fr(e)328 3452 y FE(m)395
3437 y Fr(\()p Fu(\000)p Fw(a\013)q Fr(\))p Fu(j)28 b
Fr(=)f Fu(j)p Fr(e)892 3452 y FE(q)930 3437 y Fr(\()p
Fu(\000)p Fw(b\014)6 b Fr(\))p Fu(j)28 b Fr(=)g(1)p FA(,)k(w)m(e)h(get)
g(:)257 3681 y Fq(\014)257 3741 y(\014)257 3801 y(\014)257
3860 y(\014)307 3826 y Fr(Pr)339 3885 y FE(x)428 3745
y Fq(\002)470 3826 y Fw(x)28 b Fu(2)647 3745 y Fq(\000)692
3826 y Fw(P)769 3784 y Fn(\000)p Fz(1)863 3826 y Fr(\()p
Fw(a)p Fr(\))22 b Fu(\\)h Fw(M)1195 3841 y FE(q)1233
3826 y Fr(\()p Fw(b)p Fr(\))1350 3745 y Fq(\001)1419
3826 y Fu(\000)1528 3758 y Fr(1)p 1528 3803 49 4 v 1529
3894 a Fw(q)1603 3826 y Fr(Pr)1636 3885 y FE(x)1724 3745
y Fq(\002)1766 3826 y Fw(x)28 b Fu(2)g Fw(P)2020 3784
y Fn(\000)p Fz(1)2114 3826 y Fr(\()p Fw(a)p Fr(\))2241
3745 y Fq(\003)2282 3681 y(\014)2282 3741 y(\014)2282
3801 y(\014)2282 3860 y(\014)2343 3826 y Fu(\024)2500
3758 y Fr(1)p 2458 3803 133 4 v 2458 3894 a Fw(mq)2814
3731 y Fq(X)2617 3947 y FE(\013)p Fn(2)p Fz([)p FE(m)p
Fz(])p FE(;\014)s Fn(2)p Fz([)p FE(q)r Fz(])p Fn(\000f)p
Fz(0)p Fn(g)3171 3826 y Fu(j)p Fw(S)3265 3784 y FE(m;q)3385
3826 y Fr(\()p Fw(\013)q(;)17 b(\014)6 b(;)17 b(P)d Fr(\))p
Fu(j)3295 4066 y
SDict begin H.S end
 3295 4066 a 3295 4066 a
SDict begin 17.5 H.A end
 3295 4066 a
3295 4066 a
SDict begin [ /View [/XYZ H.V] /Dest (equation.2.7) cvn H.B /DEST pdfmark
end
 3295 4066 a FA(\(2.7\))404 4300 y(The)31
b(Uniformit)m(y)d(Lemma)1444 4300 y
SDict begin H.S end
 1444 4300 a Black
FA(2.11)p Black 1617 4236 a
SDict begin H.R end
 1617 4236 a 1617 4300 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.2.11) cvn H.B /ANN pdfmark end
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4300 a 30 w FA(gets)j(pro)m(v)m(ed)h(b)m(y)f(the)g(b)s(ound)f(on)g
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b(\014)6 b(;)17 b(P)d Fr(\))p Fu(j)257 4421 y FA(pro)m(vided)49
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3425 4541 4 66 v 3429 4478 59 4 v 3429 4541 V 3487 4541
4 66 v 257 4666 a
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.13) cvn H.B /DEST pdfmark
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b(q)49 b(>)c Fr(1)f Fs(ther)-5 b(e)45 b(exists)e(a)257
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b(n)p 2158 5271 408 4 v 2158 5362 a Fr(\()p Fw(m)p Fr(2)2330
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y
SDict begin H.S end
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SDict begin 17.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (equation.2.8) cvn H.B /DEST pdfmark
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 257 266
a Black 404 573 a FA(Before)38 b(w)m(e)g(b)s(egin)f(our)h(formal)d
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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814 y(eralized)40 b(b)m(y)i(Green)f(et)g(al)f([)1329
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/Dest (cite.GRS05) cvn H.B /ANN pdfmark end
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SDict begin H.S end
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a 3134 934 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Bourgain05) cvn H.B /ANN pdfmark end
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SDict begin H.S end
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FA(27)p Black 3291 870 a
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.GRS05) cvn H.B /ANN pdfmark end
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a FA(].)404 1054 y(In)26 b(order)g(to)g(explain)f(the)h(in)m(tuition)e
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1536 y(v)-5 b(ariables)27 b(in)g Fw(I)36 b FA(in)28 b(a)f(natural)g(w)m
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Fz(\()p FE(v)r Fz(\))1983 1536 y FA(b)s(e)28 b(the)h(subfunction)f(of)g
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b(indexed)g(in)f Fw(I)39 b FA(induced)31 b(b)m(y)h(the)f(partial)d
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b(indexed)h(in)257 1777 y Fw(I)8 b FA(.)45 b(F)-8 b(or)32
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1926 y(i)p Fz(=1)2627 1897 y Fw(f)11 b Fr(\()p Fw(x)23
b Fu(\010)g Fw(y)2950 1912 y FE(i)2978 1897 y Fr(\))p
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Fw(I)8 b Fr([)p Fw(Y)21 b Fr(])45 b Fu(\022)g Fr([)p
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b(zero)g(and)h Fw(J)9 b Fr([)p Fw(Y)21 b Fr(])38 b FA(b)s(e)h(just)f
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SDict begin H.S end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.14) cvn H.B /DEST pdfmark
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2819 y(e)-5 b(ach)34 b(ve)-5 b(ctor)35 b Fw(v)d Fu(2)c(f)p
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SDict begin H.S end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Bourgain05) cvn H.B /ANN pdfmark end
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SDict begin H.R end
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a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.GRS05) cvn H.B /ANN pdfmark end
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SDict begin H.S end
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SDict begin H.R end
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2076 5388 a
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1066 y(J)1837 917 y Fq(\024)1890 1058 y Fr(e)1933 1073
y FE(m)2000 977 y Fq(\000)2045 1058 y Fw(Q)2122 1017
y FE(y)2159 993 y Fl(1)2194 1017 y FE(;:::)o(;y)2330
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Fr(\))2598 977 y Fq(\001)2643 1058 y Fr(e)2686 1073 y
FE(q)2725 1058 y Fr(\()p Fw(n)2821 1073 y FE(J)2870 1058
y Fr(\))p Fu(\002)1043 1351 y(\002)23 b Ft(E)1209 1367
y FE(u)p Fn(2f)p Fz(0)p FE(;)p Fz(1)p Fn(g)1457 1343
y Fm(n)1495 1360 y(I)1540 1211 y Fq(\024)1593 1351 y
Fr(e)1636 1366 y FE(m)1702 1270 y Fq(\000)1748 1351 y
Fw(P)1825 1300 y FE(I)5 b Fz([)p FE(Y)15 b Fz(]\()p FE(v)r
Fz(\))1811 1378 y FE(Y)2052 1351 y Fr(\()p Fw(u)p Fr(\))2184
1270 y Fq(\001)2229 1351 y Fr(e)2272 1366 y FE(q)2310
1270 y Fq(\000)2356 1351 y Fw(m)2491 1224 y FE(n)2534
1235 y Fm(I)2458 1257 y Fq(X)2473 1466 y FE(i)p Fz(=1)2618
1351 y Fw(u)2674 1366 y FE(i)2702 1270 y Fq(\001)2748
1211 y(\025)27 b(\025)3246 1351 y
SDict begin H.S end
 3246 1351 a 3246 1351
a
SDict begin 17.5 H.A end
 3246 1351 a 3246 1351 a
SDict begin [ /View [/XYZ H.V] /Dest (equation.2.10) cvn H.B /DEST
pdfmark end
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1668 y(where)33 b Fw(Q)615 1632 y FE(y)652 1608 y Fl(1)687
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y(p)s(olynomial)29 b Fw(P)14 b FA(.)404 1909 y(The)41
b(k)m(ey)g(thing)e(to)h(note)g(is)f(that)h(Observ)-5
b(ation)2284 1909 y
SDict begin H.S end
 2284 1909 a Black FA(2.14)p Black
2458 1844 a
SDict begin H.R end
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SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.2.14) cvn H.B /ANN pdfmark end
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b Fz(]\()p FE(v)r Fz(\))2899 1936 y FE(Y)3180 1909 y
FA(to)39 b(b)s(e)h(a)257 2029 y(p)s(olynomial)21 b(of)j(degree)h(at)f
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Fw(u)d FA(for)h(ev)m(ery)i(sequence)h Fw(Y)49 b Fr(=)27
b Fw(y)3030 1993 y Fz(1)3069 2029 y Fw(;)17 b(:)g(:)g(:)f(;)h(y)3340
1993 y FE(m)p Fn(\000)p Fz(1)257 2149 y FA(and)38 b(ev)m(ery)h(v)m
(ector)f Fw(v)t FA(.)57 b(Th)m(us,)40 b(the)d(inside)g(sum)g(of)44
b(\()2252 2149 y
SDict begin H.S end
 2252 2149 a Black FA(2.10)p Black 2425
2085 a
SDict begin H.R end
 2425 2085 a 2425 2149 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (equation.2.10) cvn H.B /ANN pdfmark end
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g(v)-5 b(ariable)35 b Fw(u)i FA(can)257 2270 y(b)s(e)46
b(estimated)f(using)g(our)g(inductiv)m(e)h(h)m(yp)s(othesis.)83
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y Fq(\023)1019 2894 y Fr(\(2)1106 2853 y FE(m)p Fn(\000)p
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SDict begin H.S end

3246 2894 a 3246 2894 a
SDict begin 17.5 H.A end
 3246 2894 a 3246 2894 a
SDict begin [ /View [/XYZ H.V] /Dest (equation.2.11) cvn H.B /DEST
pdfmark end
 3246
2894 a FA(\(2.11\))404 3196 y(The)34 b(rest)g(of)f(the)h(calculation)d
(pro)s(ceeds)k(exactly)e(as)h(in)f(Green)h(et.)46 b(al.)f([)3162
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SDict begin H.S end
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SDict begin H.R end
 3260
3132 a 3260 3196 a
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/Dest (cite.GRS05) cvn H.B /ANN pdfmark end
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y(rep)s(eat)33 b(it)f(here)h(for)f(the)h(sak)m(e)h(of)e(self-con)m
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h(b)s(oth)257 3437 y(sides)i(of)40 b(\()649 3437 y
SDict begin H.S end
 649
3437 a Black FA(2.11)p Black 822 3372 a
SDict begin H.R end
 822 3372 a 822
3437 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (equation.2.11) cvn H.B /ANN pdfmark end
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SDict begin H.S end
 3246 3909
a 3246 3909 a
SDict begin 17.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (equation.2.12) cvn H.B /DEST
pdfmark end
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SDict begin H.S end
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4356 a Black FA(2.8)p Black 694 4292 a
SDict begin H.R end
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4356 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (equation.2.8) cvn H.B /ANN pdfmark end
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SDict begin H.S end
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4356 a Black FA(2.13)p Black 1400 4292 a
SDict begin H.R end
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1400 4356 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.2.13) cvn H.B /ANN pdfmark end
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SDict begin H.S end
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4530 a 257 4530 a
SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (section.3) cvn H.B /DEST pdfmark
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SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (page.13) cvn H.B /DEST pdfmark
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SDict begin H.S end
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SDict begin 14.5 H.A end
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1561 a 257 1561 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.1) cvn H.B /DEST pdfmark
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SDict begin H.S end
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a
SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (subsection.3.1) cvn H.B /DEST
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1190 y FE(n)p Fz(+1)2890 1175 y Fw(x)2945 1190 y FE(n)p
Fz(+1)3121 1175 y Fr(=)39 b(0)p FA(.)63 b(By)257 1295
y(our)33 b(h)m(yp)s(othesis,)g(this)f(equation)g(on)h
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(1)f(in)f(the)i(follo)m(wing)d(w)m(a)m(y:)257 1656 y
Fw(x)312 1671 y Fz(1)380 1656 y Fr(=)e(1)p FA(,)h(for)f
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1671 y FE(i)1279 1656 y FA(is)e(set)g(to)g(the)g(v)-5
b(alue)27 b(that)f(is)h(the)g(complemen)m(t)f(of)g(its)h(v)-5
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y FE(i)1556 1777 y FA(is)g(set)h(to)f(its)f(corresp)s(onding)h(v)-5
b(alue)32 b(in)f Fw(u)p FA(.)43 b(Finally)-8 b(,)257
1897 y(extend)32 b(the)f(solutions)e(of)h Fw(g)1299 1912
y Fz(2)1338 1897 y Fw(x)1393 1912 y Fz(2)1451 1897 y
Fr(+)17 b Fu(\001)g(\001)g(\001)f Fr(+)h Fw(g)1818 1912
y FE(n)p Fz(+1)1955 1897 y Fw(x)2010 1912 y FE(n)p Fz(+1)2176
1897 y Fr(=)27 b(0)j FA(to)g(our)g(original)d(equation)j(b)m(y)257
2017 y(simply)f(\034xing)h Fw(x)888 2032 y Fz(1)956 2017
y Fr(=)d(0)j FA(to)g(obtain)f(at)h(least)g(another)g
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FE(G)p Fz(\))2667 2017 y FA(solutions.)42 b(Th)m(us,)32
b(w)m(e)257 2138 y(ha)m(v)m(e)i(at)e(least)h Fr(2)879
2102 y FE(n)p Fz(+2)p Fn(\000)p FE(s)p Fz(\()p FE(G)p
Fz(\))1245 2138 y FA(solutions)f(in)g(total,)f(pro)m(ving)h(the)h
(theorem.)p 3425 2138 4 66 v 3429 2075 59 4 v 3429 2138
V 3487 2138 4 66 v 257 2304 a
SDict begin H.S end
 257 2304 a 257 2304 a
SDict begin 14.5 H.A end
 257
2304 a 257 2304 a
SDict begin [ /View [/XYZ H.V] /Dest (subsection.3.3) cvn H.B /DEST
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b(Simple)g(F)-11 b(ourier)44 b(Analytic)i(Argumen)l(t)257
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2743 y(lo)m(wing)31 b(immediate)f(corollary:)257 2763
y
SDict begin H.S end
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SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.4) cvn H.B /DEST pdfmark
end
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a Black 191 x Fv(Corollary)44 b(3.4.)p Black 45 w Fs(L)-5
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y(m)1808 2954 y Fs(b)-5 b(e)40 b(a)g(line)-5 b(ar)39
b(map.)60 b(Then,)41 b(for)f(al)5 b(l)40 b Fw(v)h Fu(2)d
Ft(Z)3428 2918 y FE(s)3428 2978 y(m)257 3074 y Fs(such)d(that)g
Fw(K)768 3038 y Fn(L)821 3074 y Fr(\()p Fw(v)t Fr(\))f
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Fc(Z)2767 3015 y Fm(s)2767 3055 y(m)2826 3038 y Fz(\))2857
3074 y Fs(.)p Black 257 3285 a(Pr)-5 b(o)g(of.)p Black
49 w FA(Let)51 b Fu(L)59 b(\021)h(f)p Fw(`)1112 3300
y Fz(1)1151 3285 y Fw(;)17 b(:)g(:)g(:)f(;)h(`)1411 3300
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y Fr(=)257 3405 y Fw(a)308 3420 y FE(i;)p Fz(1)392 3405
y Fw(x)447 3420 y Fz(1)513 3405 y Fr(+)26 b Fu(\001)17
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3405 y Fw(x)1056 3420 y FE(n)1103 3405 y FA(.)62 b(As)39
b Fw(K)1432 3369 y Fn(L)1485 3405 y Fr(\()p Fw(v)t Fr(\))f
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3405 y FA(suc)m(h)257 3525 y(that)43 b Fw(`)520 3540
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3540 y FE(i)908 3525 y FA(.)75 b(Consider)44 b Fw(`)1469
3489 y Fn(0)1469 3550 y FE(i)1543 3525 y Fr(=)h Fw(a)1715
3489 y Fn(0)1715 3550 y FE(i;)p Fz(1)1798 3525 y Fw(x)1853
3540 y Fz(1)1923 3525 y Fr(+)29 b Fu(\001)17 b(\001)g(\001)27
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3525 y Fw(x)2475 3540 y FE(n)2523 3525 y FA(,)46 b(where)e
Fw(a)2939 3489 y Fn(0)2939 3550 y FE(i;j)3065 3525 y
Fr(=)h Fu(\000)p Fw(a)3314 3540 y FE(i;j)3438 3525 y
FA(if)257 3646 y Fw(b)298 3661 y FE(j)377 3646 y Fr(=)c(1)g
FA(and)f(otherwise)i Fw(a)1275 3610 y Fn(0)1275 3670
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Fu(L)513 3730 y Fn(0)567 3766 y Fu(\021)31 b(f)p Fw(`)766
3730 y Fn(0)766 3791 y Fz(1)806 3766 y Fw(;)17 b(:)g(:)g(:)e(;)i(`)1065
3730 y Fn(0)1065 3791 y FE(t)1095 3766 y Fu(g)p FA(.)49
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4127 y Fr(2)306 4091 y FE(n)p Fz(+1)p Fn(\000)p FE(s)p
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1289 4335 y
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a
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SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.3.4) cvn H.B /ANN pdfmark end
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SDict begin H.S end
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Black FA(3.2)p Black 1719 4391 a
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SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
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(indep)s(enden)m(t)h(w)m(ork)g(of)e(Mesh)m(ulam)g([)2837
4696 y
SDict begin H.S end
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SDict begin H.R end
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4632 a 2935 4696 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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4817 y([)284 4817 y
SDict begin H.S end
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4752 a
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Therien94) cvn H.B /ANN pdfmark end
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b(e)32 b(inlcude)e(a)h(pro)s(of)f(of)h(this,)g(using)f(simple)g(F)-8
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SDict begin H.S end
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4937 a Black FA(2.1)p Black 3468 4873 a
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4937 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (subsection.2.1) cvn H.B /ANN pdfmark end
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SDict begin H.S end
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257 5082 a
SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.5) cvn H.B /DEST pdfmark
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b(a)257 5388 y(c)-5 b(onstant.)p Black Black eop end
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SDict begin [ /View [/XYZ H.V] /Dest (page.16) cvn H.B /DEST pdfmark
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SDict begin H.R end
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y FA(supp)q Fr(\()1983 3055 y Ft(\\)1972 3078 y Fw(\025)2029
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.CGPT06) cvn H.B /ANN pdfmark end
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934 y(circuits)f(\(of)f(arbitrary)g(depth\))h(comprising)e(only)i(MOD)
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1054 y(MOD)497 1069 y FE(q)570 1054 y FA(in)e(sub-linear)f(size,)i(if)f
Fr(\()p Fw(m;)17 b(q)t Fr(\))35 b FA(are)g(co-prime.)50
b(This)36 b(signi\034can)m(tly)e(impro)m(v)m(es)257 1175
y(up)s(on)48 b(the)h(earlier)e(result)h(of)f(Smolensky)i([)1956
1175 y
SDict begin H.S end
 1956 1175 a Black FA(41)p Black 2053 1111 a
SDict begin H.R end
 2053
1111 a 2053 1175 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.Smolensky90) cvn H.B /ANN pdfmark end
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(circuits)d(need)257 1295 y Fr(\012\(log)18 b Fw(n)p
Fr(\))33 b FA(size.)43 b(F)-8 b(urther,)33 b([)1267 1295
y
SDict begin H.S end
 1267 1295 a Black FA(15)p Black 1365 1231 a
SDict begin H.R end
 1365 1231
a 1365 1295 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.CGPT06) cvn H.B /ANN pdfmark end
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i(of)f(the)h(P)m(olynomial)c(Uni-)257 1416 y(formit)m(y)g(Theorem)i
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m(h)i(circuits)257 1536 y(of)d Fs(b)-5 b(ounde)g(d)31
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(wires)g(to)g(compute)g(MOD)3171 1551 y FE(q)3209 1536
y FA(.)42 b(This,)257 1656 y(in)c(some)h(sense,)j(is)c(the)h(strongest)
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1614 y Fz(0)3176 1656 y Fr([)p Fw(m)p Fr(])h FA(cir-)257
1777 y(cuits.)404 1921 y(V)-8 b(ery)33 b(recen)m(tly)-8
b(,)34 b(Chattopadh)m(y)m(a)m(y)g(and)f(Wigderson)g([)2410
1921 y
SDict begin H.S end
 2410 1921 a Black FA(18)p Black 2508 1857 a
SDict begin H.R end
 2508
1857 a 2508 1921 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.CW09) cvn H.B /ANN pdfmark end
 2508 1921 a FA(])f(ha)m(v)m(e)i(b)s(een)g(able)e(to)
g(sig-)257 2041 y(ni\034can)m(tly)f(impro)m(v)m(e)f(Theorem)1463
2041 y
SDict begin H.S end
 1463 2041 a Black FA(3.9)p Black 1588 1977 a
SDict begin H.R end
 1588
1977 a 1588 2041 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.3.9) cvn H.B /ANN pdfmark end
 1588 2041 a 31 w FA(for)g(the)h(case)h(of)e(linear)f
(systems)k(under)e(the)g(con-)257 2162 y(dition)g(that)h
Fw(m)h FA(is)f(precisely)h(a)f(pro)s(duct)h(of)f(t)m(w)m(o)h(primes.)
257 2181 y
SDict begin H.S end
 257 2181 a 257 2181 a
SDict begin 14.5 H.A end
 257 2181 a 257 2181
a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.11) cvn H.B /DEST pdfmark
end
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b(et)44 b Fw(m;)17 b(q)46 b Fs(b)-5 b(e)43 b(c)-5 b(oprime)42
b(p)-5 b(ositive)257 2451 y(inte)g(gers,)50 b(with)e
Fw(m)j Fr(=)h Fw(p)1203 2466 y Fz(1)1242 2451 y Fw(p)1291
2466 y Fz(2)1378 2451 y Fs(and)47 b(e)-5 b(ach)47 b Fw(p)1861
2466 y FE(i)1937 2451 y Fs(is)g(a)g(prime.)83 b(Ther)-5
b(e)46 b(exists)i(a)f(p)-5 b(ositive)257 2571 y(c)g(onstant)37
b Fw(\015)g Fr(=)32 b Fw(\015)5 b Fr(\()p Fw(m;)17 b(q)t
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b(lowing)36 b(holds:)49 b(let)37 b Fu(L)32 b Fr(=)g Fu(f)p
Fw(`)3117 2586 y Fz(1)3156 2571 y Fw(;)17 b(:)g(:)g(:)f(;)h(`)3416
2586 y FE(t)3445 2571 y Fu(g)257 2691 y Fs(b)-5 b(e)27
b(a)h Fw(n)p Fs(-variate)f(line)-5 b(ar)27 b(system)g(over)g
Ft(Z)1713 2706 y FE(m)1780 2691 y Fs(,)i(with)e(ac)-5
b(c)g(epting)26 b(sets)i Fu(f)p Fw(A)2764 2706 y Fz(1)2803
2691 y Fw(;)17 b(:)g(:)g(:)f(;)h(A)3095 2706 y FE(t)3124
2691 y Fu(g)p Fs(.)42 b(Then,)1344 2991 y(disc)1508 3006
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y Fn(L)1703 2910 y Fq(\001)1793 2991 y Fu(\024)j Fs(exp)2054
2910 y Fq(\000)2122 2991 y Fu(\000)23 b Fw(\015)5 b(n)2336
2910 y Fq(\001)2382 2991 y Fw(:)3295 2991 y
SDict begin H.S end
 3295 2991
a 3295 2991 a
SDict begin 17.5 H.A end
 3295 2991 a 3295 2991 a
SDict begin [ /View [/XYZ H.V] /Dest (equation.3.3) cvn H.B /DEST pdfmark
end
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3170 y(An)30 b(in)m(teresting)g(thing)f(to)g(note)h(is)f(that)h(the)g
(constan)m(t)h Fw(\015)j FA(in)29 b(equation)h(\()2964
3170 y
SDict begin H.S end
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SDict begin H.R end
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3106 a 3088 3170 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (equation.3.3) cvn H.B /ANN pdfmark end
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(system.)43 b(The)32 b(argumen)m(t)e(of)g([)2735 3290
y
SDict begin H.S end
 2735 3290 a Black FA(18)p Black 2832 3226 a
SDict begin H.R end
 2832 3226
a 2832 3290 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.CW09) cvn H.B /ANN pdfmark end
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3411 y(and)35 b(com)m(bines)g(ideas)g(of)f(using)g(exp)s(onen)m(tial)g
(sums)h(from)f([)2521 3411 y
SDict begin H.S end
 2521 3411 a Black FA(15)p
Black 2619 3346 a
SDict begin H.R end
 2619 3346 a 2619 3411 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.CGPT06) cvn H.B /ANN pdfmark end
 2619 3411 a
FA(],)h(estimates)f(of)h(Bour-)257 3531 y(gain)29 b(\(Lemma)849
3531 y
SDict begin H.S end
 849 3531 a Black FA(2.13)p Black 1023 3467 a
SDict begin H.R end
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3467 a 1023 3531 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.2.13) cvn H.B /ANN pdfmark end
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(the)i(notion)e(of)g(matrix)f(rigidit)m(y)g(from)h(the)257
3651 y(ingenious)35 b(w)m(ork)i(of)e(Grigoriev)e(and)j(Razb)s(oro)m(v)g
([)2140 3651 y
SDict begin H.S end
 2140 3651 a Black FA(28)p Black 2237 3587
a
SDict begin H.R end
 2237 3587 a 2237 3651 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.GR00) cvn H.B /ANN pdfmark end
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(not)j(allo)m(w)f(us)h(to)g(co)m(v)m(er)h(the)g(en)m(tire)f(argumen)m
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(main)e(ideas)h(in)m(v)m(olv)m(ed)h(in)f(pro)m(ving)g(the)h(T)-8
b(w)m(o-Prime)26 b(Uniformit)m(y)257 4013 y(Theorem.)257
4146 y
SDict begin H.S end
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SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.3.4.1) cvn H.B
/DEST pdfmark end
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4146 a 120 x Fv(3.4.1)113 b(Singleton)36 b(A)m(ccepting)g(Sets)257
4451 y FA(T)-8 b(o)39 b(b)s(egin)e(with,)i(let)f(us)h(assume)f(that)g
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Fw(A)1282 4586 y FE(i)1338 4571 y Fu(\021)28 b(f)p Fr(0)p
Fu(g)p FA(.)43 b(Then,)34 b(as)f(b)s(efore,)f(one)h(can)g(write)274
4871 y Fr(Pr)306 4931 y FE(x)395 4790 y Fq(\002)436 4871
y Fw(f)484 4886 y Fn(L)537 4871 y Fr(\()p Fw(x)p Fr(\))28
b(=)g(1)22 b Fu(^)g Fw(x)28 b Fu(2)g Fw(M)1230 4886 y
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5155 y Fr(=)g Ft(E)455 5171 y FE(x)p Fn(2f)p Fz(0)p FE(;)p
Fz(1)p Fn(g)702 5152 y Fm(n)749 5015 y Fq(\024)16 b(\032)961
5031 y FE(t)910 5061 y Fq(Y)916 5270 y FE(i)p Fz(=1)1054
5015 y Fq(\022)1155 5088 y Fr(1)p 1137 5132 86 4 v 1137
5223 a Fw(m)1249 5031 y FE(m)p Fn(\000)p Fz(1)1253 5061
y Fq(X)1257 5270 y FE(\013)p Fz(=0)1418 5155 y Fr(e)1461
5170 y FE(m)1528 5074 y Fq(\000)1574 5155 y Fw(\013)q
Fr(\()p Fw(`)1716 5170 y FE(i)1743 5155 y Fr(\()p Fw(x)p
Fr(\))23 b Fu(\000)f Fw(a)2047 5170 y FE(i)2076 5155
y Fr(\))2114 5074 y Fq(\001)2159 5015 y(\023)q(\033)o(\022)2391
5088 y Fr(1)p 2391 5132 49 4 v 2392 5223 a Fw(q)2476
5027 y FE(q)r Fn(\000)p Fz(1)2466 5061 y Fq(X)2471 5273
y FE(\014)s Fz(=0)2627 5155 y Fr(e)2670 5170 y FE(q)2708
5074 y Fq(\000)2754 5155 y Fw(\014)6 b Fr(\()p Fw(x)2908
5170 y Fz(1)2969 5155 y Fr(+)22 b Fu(\001)17 b(\001)g(\001)k
Fr(+)h Fw(x)3359 5170 y FE(n)3428 5155 y Fu(\000)h Fw(b)p
Fr(\))3607 5074 y Fq(\001)3653 5015 y(\023)17 b(\025)3295
5388 y
SDict begin H.S end
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SDict begin 17.5 H.A end
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a
SDict begin [ /View [/XYZ H.V] /Dest (equation.3.4) cvn H.B /DEST pdfmark
end
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SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (page.19) cvn H.B /DEST pdfmark
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a Black 404 573 a FA(Mimic)m(king)26 b(argumen)m(ts)j(used)h(in)e(the)i
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SDict begin H.S end
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FA(2.4)p Black 899 629 a
SDict begin H.R end
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SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (equation.2.4) cvn H.B /ANN pdfmark end
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a FA(\))i(to)f(\()1127 693 y
SDict begin H.S end
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Black 1252 629 a
SDict begin H.R end
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SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (equation.2.7) cvn H.B /ANN pdfmark end
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(obtain,)582 1125 y(disc)745 1140 y FE(q)783 1044 y Fq(\000)829
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1044 y Fq(\000)2401 1125 y Fw(b)p Fr(\()p Fw(x)2535 1140
y Fz(1)2598 1125 y Fr(+)22 b Fu(\001)17 b(\001)g(\001)j
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y Fq(\001)3118 985 y(\025)257 1451 y FA(where,)41 b(eac)m(h)f
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Fw(r)981 1587 y FE(j)1018 1572 y Fr(\()p Fw(x)p Fr(\))e(=)f
Fw(a)1335 1587 y FE(j;)p Fz(1)1423 1572 y Fw(x)1478 1587
y Fz(1)1541 1572 y Fr(+)23 b Fu(\001)17 b(\001)g(\001)k
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1587 y FE(n)2160 1572 y FA(,)34 b(w)m(e)h(can)f(separate)h(v)-5
b(ariables)32 b(and)257 1692 y(obtain)257 1830 y Fq(\014)257
1890 y(\014)257 1949 y(\014)257 2009 y(\014)291 1974
y Ft(E)357 1990 y FE(x)p Fn(2f)p Fz(0)p FE(;)p Fz(1)p
Fn(g)604 1971 y Fm(n)651 1834 y Fq(\024)704 1974 y Fr(e)747
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y FE(j)940 1974 y Fr(\()p Fw(x)p Fr(\))1071 1894 y Fq(\001)1117
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1974 y Fw(b)p Fr(\()p Fw(x)1378 1989 y Fz(1)1440 1974
y Fr(+)22 b Fu(\001)17 b(\001)g(\001)k Fr(+)h Fw(x)1830
1989 y FE(n)1877 1974 y Fr(\))1915 1894 y Fq(\001)1961
1834 y(\025)2013 1830 y(\014)2013 1890 y(\014)2013 1949
y(\014)2013 2009 y(\014)2074 1974 y Fr(=)2220 1850 y
FE(n)2178 1880 y Fq(Y)2184 2090 y FE(i)p Fz(=1)2322 1830
y Fq(\014)2322 1890 y(\014)2322 1949 y(\014)2322 2009
y(\014)2355 1974 y Ft(E)2421 1990 y FE(x)2461 2000 y
Fm(i)2487 1990 y Fn(2f)p Fz(0)p FE(;)p Fz(1)p Fn(g)2699
1834 y Fq(\024)2752 1974 y Fr(e)2795 1989 y FE(m)2862
1894 y Fq(\000)2907 1974 y Fw(a)2958 1989 y FE(j;i)3035
1974 y Fw(x)3090 1989 y FE(i)3118 1894 y Fq(\001)3164
1974 y Fr(e)3207 1989 y FE(q)3245 1894 y Fq(\000)3291
1974 y Fw(bx)3387 1989 y FE(i)3416 1894 y Fq(\001)3462
1834 y(\025)3514 1830 y(\014)3514 1890 y(\014)3514 1949
y(\014)3514 2009 y(\014)1696 2217 y Fu(\024)29 b FA(exp)1951
2137 y Fq(\000)2018 2217 y Fu(\000)23 b Fw(\013)q(n)2239
2137 y Fq(\001)257 2449 y FA(for)35 b(some)h Fr(0)c Fw(<)h(\013)g(<)g
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b(v)m(erify)h(that)g(it)e(pro)m(v)m(es)j(Theorem)1657
3656 y
SDict begin H.S end
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SDict begin H.R end
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3592 a 1781 3656 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.3.9) cvn H.B /ANN pdfmark end
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(of)f(linear)f(systems.)257 3676 y
SDict begin H.S end
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a
SDict begin 14.5 H.A end
 257 3676 a 257 3676 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.12) cvn H.B /DEST pdfmark
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871 4907 a
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